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\title{Immune System Overview}
\author{Ricardo Cruz}

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\section{Introduction}

The greatgrandfather of the immune system is thought to have been a system of self-replicating particles from the primordial soup that managed to form bodies through self-self binding. The immune system is that which distinguishes between self and non-self molecules, in order to keep our body whole. It stops foreign molecules at entrance, or expels them afterwards.

Human immune system is usually divided into innate and adaptive components. The \textbf{innate system} is common to all animals (including plants and fungi), and consists in great part of physical and chemical barriers; the gastric acid of the stomach would be such an example, reducing chances of food contamination by killing worm parasites at entrance. But more than that, the innate system encompasses \glspl{white blood cell} that identify and neutralize the historically most common invaders by recognizing patterns of molecules characteristic of general categories of \glspl{pathogen}. The big selling points of the innate system are its fast response (in the order of minutes), and not requiring previous exposure to the \gls{pathogen}.\glsadd{innate system}

On the other hand, the \textbf{adaptive system} is a ``recent'' invention, only available to vertebrates. It identifies molecules (usually specific proteins) found only in a specific strain of the \gls{pathogen}. It requires exposure and is slower --- in the order of weeks for a first exposure, and in the order of days afterwards. It combines synergies and works in harmony with the innate system.\glsadd{adaptive system}

\smallskip
\begin{minipage}{0.45\linewidth}
\belongs{innate}{
	\glspl{macrophage} \\ \glspl{neutrophils} \\ \glspl{dendritic} \\ \glspl{NK}}
\end{minipage}
\begin{minipage}{0.45\linewidth}
\belongs{adaptive}{
	\glspl{thcell} \\ \glspl{tccell} \\ \glspl{bcell}}
\end{minipage}
\smallskip

The immune system is an interwinded, complex system that is not yet fully charted. It is not perfect, and is the target of much research. For our study, we are interested in how HIV overpowers and shuts down the immune system, leaving it vulnerable to opportunistic invaders. Other interesting areas of research include: why the immune system fights transplanted tissues and organs; does not kill off out of control cells (cancer); and the misdirection of defenses, such as over-the-top immune response against non-pathogens (allergy) or when the immune system identifies self tissues as non-self and attacks them (autoimmune disease).

\section{Summary of immune cells}
\label{immunecells}

This is a general listing of types of cells that constitute our immune defenses. After centrifugation of a blood sample, these are the cells that can be found in the liquified segment layered between the red blood cells (that carry oxygen) and blood plasma (the blood fluid). After centrifugation, they have a white tonality which is why they are known as \glspl{white blood cell}. They make up less than 1\% of total blood volume of an adult. \citep[Table 22-1]{Alberts2002}

The various cells can be divided by their various functions, some of which overlap:

\medskip
\noindent
\begin{minipage}{0.30\linewidth}
\belongs{antigen\\presenting\\cells}{
	\glspl{macrophage} \\ \glspl{dendritic} \\ \glspl{bcell}}
\end{minipage}
\hfill
\begin{minipage}{0.30\linewidth}
\belongs{humoral\\response}{
	\glspl{bcell}}
\end{minipage}
\hfill
\begin{minipage}{0.30\linewidth}
\belongs{cell-mediated\\immunity}{
	phagocytes \\ \glspl{tccell} \\ release of \glspl{cytokine}}
\end{minipage}
\medskip

\vspace{-\topsep}
\begin{itemize}
\item \textbf{\Acrfull{APC}} \glsdesc*{APCg}
\item \textbf{Humoral response}, also known as antibody-mediated beta cellularis immunity, refers to the production of antibodies, and the process leading to it. Antibodies are then expelled from the body through several fluids (humors) such as mucus and excrement, which were thought to be related to humor states of mind in ancient medicine.
\item \textbf{Cell mediated immunity} is the response that does not involve antibodies, but rather phagocytes directly ingesting foreign molecules, \glspl{tccell} destroying infected cells, and the release of various communication \glspl{cytokine}.
\end{itemize}
\vspace{-\topsep}

Following is a listing of major classes of immune cells, together with their \emph{concentration} per blood sample of \glspl{white blood cell} to give an idea of their overall importance, and whether they are usually \emph{modeled} in HIV dynamics. This last point will be better explored in the next chapter.

\newcommand{\agent}[3] {  % \agent
\raisebox{-0.85\height}{\includegraphics[width=6em]{chp1/#1.jpg}} & \textbf{\Gls{#1}} \glsdesc*{#1} & #2 \% & #3 \\
}

\noindent
\begin{longtable}{lp{28em}cc}
\textbf{Cell} & \textbf{Summary} & \textbf{Concentration} & \textbf{Modeled?} \\\midrule
\endhead
\agent{neutrophils}{?}{no}
\raisebox{-0.85\height}{\includegraphics[width=6em]{chp1/mast.jpg}} & \textbf{Mast cell} \& other granulocytes (besides the neutrophils) are not numerous and reside in mucosal tissue. They play a role in both parasitic infections and allergies. & & no \\
\agent{macrophage}{?}{sometimes}
\agent{dendritic}{?}{often}
\agent{bcell}{?}{often}
\agent{thcell}{?}{always}
\agent{tccell}{?}{often}
\agent{NK}{?}{no}
& \textbf{Other white blood cells} include the natural killer T cells (not to be confused with the \glspl{tccell} or the \glspl{NK}) --- they make up of 0.1\% of the blood and will not be the subject of exposition here. & & no \\
\hline
\end{longtable}

And last but not least, the utmost ``agent'' studied by the models at hand is the molecule:

\noindent
\begin{longtable}{lp{28em}cc}
\textbf{Molecule} & \textbf{Summary} & \textbf{Concentration} & \textbf{Modeled?} \\\midrule
\endhead
\agent{HIV}{}{always}
\hline
\end{longtable}






\section{Antigen Presenting Cells}
\label{apc}


\Acrfull{APC} \glsdesc*{APCg} These cells incorporate an antigen (such as an opsonized pathogen) and codify it for display in proteins on their surface called class II \gls{MHC}. They serve to activate and stimulate \glspl{lymphocyte}, triggering immune response, as explain in section \ref{lymphocyte}.

While circulating, T cells are restimulated by \glspl{macrophage}. These \glsdesc*{macrophage}

But the most important of \gls{APC} cells are \textbf{\glspl{dendritic}} as studied in the following section.




\subsection{Macrophages}
\label{macrophage}

One type of \acrfull{APC} are \glspl{macrophage}. These cells lack mobility. They have been considered crucial for HIV dynamics because they feature CD4 proteins in their surface (such as \glspl{thcell}), and the fact that primary infection seems largely restricted to relatively homogeneous M-tropic HIV virus. The several strains of HIV virus can be divided into the following two categories. They are known as HIV tropism:

\vspace{-\topsep}
\begin{itemize}
\item \textbf{M-tropic} virus mostly uses beta-chemokine receptor CCR5 for entry, and can infects both \glspl{thcell} and \glspl{macrophage}.
\item \textbf{T-tropic} virus mostly uses alpha-chemokine receptor CXCR4 for entry, and infects primarily \glspl{thcell}, but may also infect \glspl{macrophage}.
\end{itemize}
\vspace{-\topsep}

The fact that M-tropic HIV virus is prevalent during primary infection seems to imply an important role for \glspl{macrophage}.

Carriers of ``delta 32'' mutation are resistant to M-tropic strains of HIV-1 infection because of a defection in CCR5 expression. The Berlin patient --- the only known patient to apparently have been cured of HIV-1 --- received bone marrow transplant from a person suffering of this mutation.







\subsection{Dentric cells}
\label{dendritic}

\Glspl{dendritic} are ``sentinel'' cells which take up positions beneath the barriers of epithelial cells. When the tissue in which they reside becomes a battle site, the \glspl{cytokine} that are produced activate the \gls{dendritic} and make it travel to the closest \gls{lymph node}, in order to present antigens to the lymphocytes there. The antigens it captures are advertised in the class I MHC molecules it features on its surface, which it upregulates after being activated. Its lifetime after activation is small so that following \glspl{dendritic} may report more updated information on the infection, though its lifetime is prolonged when it engages with the helper T cell. \Glspl{dendritic} are imprinted with the local infection cytokines, which means that, has they come to \glspl{lymph node}, other cells know the source of the infection. The \gls{dendritic} is not to be confused with the follicular \glspl{dendritic} which make up the \gls{lymph node} structure.

Dendritic cells found in epidermal skin and mucus are known as ``langerhans cells''. These cells are one of the first contact points of HIV-1 since they can be found in the foreskin of the penis, as well as in vaginal and oral mucosa. Research publications agree these dendritic cells play a role of important magnitude in HIV-1 infection. However, there is disagreement on the direction of the contribution. Some argue that HIV directly infects \glspl{thcell} in the foreskin or in mucus, and act as a barrier for HIV propagation because once captured by them the virus quickly degrades. Others however argue these dendritic cells are the ones responsible for taking the virus to \glspl{thcell}. And, while \glspl{dendritic} can be infected \emph{in vitro}, there is actual dispute on whether they are actually infected \emph{in vivo}, because no evidence of such has been found in infected humans, though monkey experiments do show infected \glspl{dendritic}. \citep{Wilflingseder2005} While a meta study on the amount of papers defending each side of the argument, I am going to avoid taking a position by not explaining first contact in the model and assume that time starts counting from the first \gls{thcell} infection, as most HIV dynamical models seem to assume. Some older models like ones based on ImmSim did assume dendritic participation in fomenting disease (see section \ref{immsim} from the following chapter).

The initial infection contact of HIV is still contested, but it appears that \glspl{dendritic} are one of the first cells to be infected and transfer the virus to \glspl{thcell}. Such transfer may occur locally in inflamed mucosa or after \glspl{dendritic} have matured and migrated to local \glspl{lymph node}. The probability of infection of immature \glspl{dendritic} is low, and happens through C-type lectin receptors (CLRs). The most common CLR in both \gls{dendritic} and \gls{macrophage} is DC-SIGN. The low probability leads to a negative selection of R5 strain (T-tropic) HIV, and viruses that persist beyond the 24h travel time of the \gls{dendritic} to the \gls{lymph node}. \citep{Turville2003} As \glspl{dendritic} mature, the ``quasi-monopoly'' of CCR5 over CXCR-4 receptors is challenged, as CCR5 receptores are down-regulated. Infection of \glspl{dendritic} is also of lower productivity than that of \glspl{thcell} and of \glspl{macrophage}; and it has been suggested infection of \glspl{dendritic} may not be necessary, and these cells act only as vehicles to transport the virus to the \gls{lymph node}. However, this point is still contended since HIV progressively degrades during the 24h travel time. It is probably safer to assume \gls{dendritic} infection as the vehicle for transport and subsequent creation of the virus in the \gls{lymph node}.

It should be noted that lymph nodes are made up of a mesh of ``follicular dendritic cells'' that, despite the name, are completely unrelated to ``dendritic cells''. They are derived from a different lineage.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
Travels after & 6 hours into the battle, capturing antigens in the hitherto & \citep{Sompayrac2012} \\
Lifetime & 1 week after reaching the lymph node & \citep{Sompayrac2012} \\
Interactions & it is typically visited 1,000 per hour within the lymph node & \citep{Sompayrac2012} \\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}




\section{Lymphocytes}
\label{lymphocyte}

By definition, lymphocyte refers to cells: T, B and Natural Killer. The name comes from the fact these are the cells mainly found at the \gls{lymph node}. There is no other biological basis for the definition.

\noindent
\belongs{lymphocytes}{
helper T cells \\
killer T cells \\
B cells \\
natural killers}

\begin{wrapfigure}{r}{8em}
\begin{tikzpicture}
\node [state] (naive) {naive};
\node [state, above right=of naive] (effector) {effector};
\node [state, below right=of naive] (memory) {memory};

\path [arrow] (naive) -- (effector);
\path [arrow] (naive) -- (memory);
\end{tikzpicture}
\end{wrapfigure}

Some properties are common to all \textbf{lymphocytes}, which are the main defense coordinators and responders, operating from the \gls{lymph node} --- B and T cells will be described afterwards in more detail. These cells proliferate when exposed to an antigen, forming effector and memory lymphocytes. Effector lymphocytes act to eliminate the antigen; either by releasing antibodies (B cells), cytotoxic granules (killer T cells), or by signaling to other cells (helper T cells). Memory lymphocytes remain in the system and circulate for an extended period ready to respond to the same antigen upon future exposure.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Class} & \textbf{Function} & \textbf{Source} \\\midrule
Helper T cells & 6 hours into the battle, capturing antigens in the hitherto & \citep{Sompayrac2012} \\
Killer T cells & 1 week after reaching the lymph node & \citep{Sompayrac2012} \\
Interactions & it is typically visited 1,000 per hour within the lymph node & \citep{Sompayrac2012} \\
\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}






\subsection{B cells}
\label{bcell}

B cells make up the humoral response of the system, by producing antibodies that bind against the attacker. A molecule that binds to an antibody is called an antigen.

The antibody protein is just like the \gls{BCR}, except that it lacks the protein sequences at the tip of the heavy chain which anchors the \gls{BCR} to the surface of the cell; so antibodies are the receptors that are released from the cell. They travel through the blood stream and bind to epitopes of antigens, neutralizing them. It is not known exactly how the BCR protein is sequenced. We do know that all \glspl{BCR} of a mature B cell share the same protein sequence. It is the result of a mix and match strategy, so each B cell has their own receptor specification; there is such a multitude of possible receptors that collectively B cells could recognize virtually any organic molecule. B cells mature in the bone marrow. (T cells mature in the thymus.) They are found primarily in \glspl{lymph node}, though they also circulate.

B cells are not easily triggered to produce antibodies. The BCR (outside the surface) recognizes the antigen but the activation signal is only sent by Ig$\alpha$ and Ig$\beta$ proteins which are located inside the cell. Since these two parts of the cell are disjoint, the activation signal is only sent when many BCR cluster at the surface of the cell --- they are then said to have been \emph{crosslinked}, although they are not actually linked --- which can happen when several BCR bind to epitopes on antigens that are clumped together. The B cell also has a receptor protein that recognizes an opsonized antigen, speeding up the process of B cell activation and reducing the number of BCR that must be clustered. An antigen is said to be opsonized when it is disable by an antibody --- meaning the cell knows the antigen has already been recognized as a threat previously. Before being activated, a B cell is said to be naive or virgin. Afterwards they are said to be experienced.

Crosslink is not enough to fully activate the B cell; a further signal is necessary. This co-stimulatory signal may be supplied by an activated helper T cell when a helper T cells surface protein called CD40L plugs into the CD40 protein at the surface of the B cell. The B cell is then activated, if it was crosslinked. In certain cases the activation signal may be T-cell independent; that can happen with certain antigens that have repeated epitopes which can cosslink a ton of B cell receptors, and the B cell recognizes molecular patterns characteristic of certain bacteria and parasites --- we are only concerned about T-cell dependent activation for our case study.

Once B cells have been acivated and have proliferated to build up their numbers, there comes the next stage of B cell life cycle: maturation. Maturation can be divided into three steps, not necessarily sequential: \emph{class switching} in which the B cell chooses what class of antibody to produce, \emph{somatic hypermutation} in which the rearranged genes of the BCR undergo mutation and selection that may further increase affinity to the antigen.

\noindent
\begin{center}
\begin{tikzpicture}
\node [state] (naive) {naive};
\node [state, right=of naive] (activated) {activated};
\node [state, right=2.5em of activated] (switching) {class switching};
\node [state, below=of switching] (hypermutation) {somatic hypermutation};
\node [basenode, above=of switching, left] (maturation) {maturation};
\node [state, right=2em of hypermutation] (memory) {memory};
\node [state, above=of memory] (plasma) {plasma};

\node[fit=(switching) (hypermutation)] (maturation-group) {};
\node[state, inner sep=0.1em, outer sep=0, fit=(maturation) (maturation-group)] {};

\path [arrow] (naive) -- (activated);
\path [arrow] (activated) -- (switching);
\path [arrow] (switching) -- (hypermutation);
\path [arrow] (hypermutation) -- (plasma);
\path [arrow] (hypermutation) -- (memory);
\end{tikzpicture}
\end{center}

During \textbf{class switching}, the B cell comes to produce different antibodies. What influences class switching are the cytokines of the environment. The cytokines are usually produced by the helper T cells while activating the B cell; helper T cells in turn know which cytokines to produce based on the source cytokines imprinted in dendritic cells. The antigen binding region is always the same, but the F\textsubscript{C} region of the heavy chain differs; meaning they are better suited for different tasks. For example, IgA antibodies are good at collecting \glspl{pathogen} together in clumps that are swept out of the body with mucus or feces --- in fact, rejected bacteria account for  30\% of normal fecal matter. The default class of antibodies are IgM, which are common to lower vertebrates as well. IgG is particularly suited for viruses, and are the ones produced in response to HIV.

Ordinary mutation rates are pretty low --- about one base per 100 million bases per DNA replication cycle. But, during \textbf{somatic hypermutation}, B cells may reach mutation rates as high as one mutated base per 1,000 bases per generation. For B cells to continue to proliferate, and build up numbers, they must be continually re-stimulated, by binding to its cognate antigen; this results in a selection pressure for higher affinity B cells, since those that respond more easily to the antigen would also proliferate more easily. Current models (using strings and Hamming distance) are incapable of capturing the richfulness of such phenomenon. Somatic hypermutation is also controlled by helper T cells cytokines.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
Production & one billion per day & \citep{Sompayrac2012} \\
Cell BCR count & 100,000 & \citep{Sompayrac2012} \\
Opsonized speedup & 100 fold & \citep{Sompayrac2012} \\
IgG lifetime & 3 weeks & \citep{Sompayrac2012} \\
Antibody production & 2000 antibodies per second  & \citep{Sompayrac2012} \\
MHC II presentation & less than half an hour after exposure & \citep{Sompayrac2012} \\
\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}



\subsection{Helper T cells}
\label{thcell}

Helper T cells act as the decision maker and informer of B cells, after collecting information from a dendritic cell.

Like B cells, for a helper T cell to be activated, its receptors must recognize their cognate antigen, as displayed by class II MHC molecules on the surface of activated dendritic cells, thus ensuing crosslink. Furthermore a co-stimulatory signal is required from the dendritic cell; this signal involves protein B7 located on the surface of the dendritic cell to plug to a protein called CD28 on the surface of the T cell. Resting dendritic cells also feature B7 proteins and low levels of MHC molecules, but only when they become activated do they produce B7 and MHC in quantities enough to stimulate virgin T cells. Three types of antigen presenting cells have been identified: activated dendritic cells, activated macrophages, and activated B cells. Like the other \glspl{white blood cell}, antigen presenting cells are continuously being replenished. It is estimated that B cells (as antigen presenting cells) are 100 to 10,000-fold better at stimulating helper T cells. Notice that dendritic cells continue being the main antigen presenting cells, because they are the first to signal the infection.

Helper T cells differentiate to produce different cytokines, depending on the cytokine produced by the dendritic cell as it arrives the \gls{lymph node} --- which in turn depends on the source of the dendritic cell, as previously seen. There are three major subsets of helper T cells: Th1, Th2, and Th17 (usually they are a mixture). Different helper T cells produce different cytokines which instruct B cell about which antibodies to produce after class switching. We are mostly concerned with Th1 cells. Virgin T cells mature as Th1 cells when presented with the IL-12 cytokine by the dendritic cell. They will in turn secrete TNF cytokines that help activate macrophages and natural killers. They also produce IFN-$\gamma$ which influences B cells to class switch to produce IgG3 antibodies. Furthermore, they also produce IL-2, which is a cytokine that commands growth, and thus stimulates proliferation of killer T cells, natural killers, and Th1 cells themselves. Interestingly, there are some helper T cells, so-called Th0, that never choose a function and can perform the action of all previous helper T cell types.

HIV targets the CD4 protein, which is mainly found in helper T cells --- this is also why helper T cells are sometimes called CD4\textsuperscript{+} T cells. In HIV dynamics, only helper T cell infection is usually considered, given they make up the majority of CD4\textsuperscript{+} cells. The CD nomenclature refers to clusters of proteins primarily found at the surface of cells. The number in the CD nomenclature refers not to any specific trait or configuration of the protein, but merely to the chronological order in which they were discovered. Because cells have several proteins in its surface, they are sometimes referred with a ``+'' suffix indicating the cell expresses the protein, or a ``-'' suffix indicating that it does not.




\subsection{Killer T cells}
\label{tccell}

Killer T cells destroy cells if it binds to its class I MHC molecules, which are located within the interior of the cell and indicate the presence of an antigen.

When a virus infects a cell and the cell becomes productively infected, those molecules allow the killer T cells to find proteins produced by the antigen within the cell it has infected. Viral epitopes cannot hide from killer T cells. When virus-infected cells die by apoptosis, the DNA of unassembled viruses is destroyed along with the target cell's DNA; everything is then disposed of by macrophages.

Interestingly, killer T cells are also responsible for the destruction of misbehaving cells, such as cancerous cells. They are therefore an important part of our immune system, and are often times included in immune models. It is still a mystery how killer T cells become activated. But it is thought that without the help of a helper T cell, killer T cells do not last long.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
Proliferation & ~10,000 during first week & \citep{Sompayrac2012} \\
\hline
\end{tabularx}
\rowcolors{1}{}{}
\end{center}




\subsection{Natural killers}
\label{NK}

Natural killers complement killer T cells by destroying cells that fail to produce class I MHC molecules; those  may be ``stressed'' cells, or may be hiding a virus.

Two classes of Major Histocompatibility Complex (MHC) exist. \textbf{MHC I} may be found in the interior of all nucleated cells, and is inspected by killer T cells. On the other hand, \textbf{MHC II} molecules are found in any antigen presenting cell (from the dendritic cells to the B cell) and are external to the cell, advertising the antigen to helper T cells. Hence why cells featuring MHC II are called antigen presenting cells. The first MHC to be discovered was class I, and it was discovered in relation to transplant rejections, hence its name.




\section{Thymus \& Bone Marrow (distributed generator)}
\label{thymus}

Thymus and the bone marrow are primary lymphoid organs producing T cells (thymus) and B cells (the blood marrow; as well as dendritic cells and all other myeloid cells). These primary organs are not usually explicit in the model, and their function is modeled exogenously, through a continuous and constant or variable production of \glspl{white blood cell}. When the production of cells is variable, --- meaning the production rates of new cells is impacted by the disease --- it is said these organs are implicitly modeled.

Each newborn (naive) cell is different from each other, in order to react to different molecules. Through the daily production of \glspl{white blood cell} in the order of the billions, this garantees our immune system has somewhere a cell that can react to virtually any molecule in the universe. They are ``tested'' against self-molecules of the body, and, if they react, they are not allowed into the system. This process called of anti-selection serves as the basis of the self-nonself recognization system.

Both helper and killer T cells are continuously produced and trained in the thymus. Like the blood marrow for B cells, the thymus produces T cells with different specificities that can respond to different organic molecules. There is a process of negative selection so that only T cells that do not respond to self molecules are allowed to exit the Thymus. They must be activated before they can perform their respective function. In agent-based modeling, specificity between T cells and their cognate antigen is given by a string and an affinity potential by a Hamming distance, as seen previously relatively to B cells. Nowak introduced viral strains in differential equation models, there is no concept of binding affinities.





\section{Specificity}
\label{specificity}

Variance in the shapes of proteins on the cells' surface is what allow the immune system to recognize non-self invaders, giving rise to different degrees of cellular \textbf{\gls{specificity}}. \Glspl{white blood cell} have different degrees of affinity between each other; also known as the shape space as proposed by \citet{Perelson1979}.

Given the virtual impossibility of realistically simulating molecular binding, a common approach, that started with ImmSim, is to take specificity of the receptor or paratope of each cell or virion as a 8-bit string (a byte). Given a Hamming complementarity distance between the strings of two agents, a binding probability is calculated based on an affinity potential function. \citet{Lin2010} use a simple binding probability function given by $P=p_0 \exp(-\alpha h_{\text{i,j}})$, where $h_{\text{i,j}}$ is the Hamming distance between the specificity of the two interacting proteins. Like Nowak, they use a 10-bit string, giving rise to $2^{10}=1024$ different strains of binding molecules. They simulate a mutation by randomly flipping one bit value of the string. Notice that specificity is not only simulated between BCRs/antibodies and antigens, but also between B cells and helper T cells, and etc.

Most \glspl{ABM} represent this \gls{specificity} as a binary string, and the \gls{affinity} between two binding sites is thus given by their complementarities. While seemingly irrealistic, it is not so far-fetched given that a \gls{point mutation} may suffice to change the specificity of both the transcription activation protein and its binding site. \citep{Retallack1993} \citep{Smith2004}

\begin{center}
\begin{tikzpicture}
\begin{scope}
\clip (1em,1.80em) rectangle (9em,8.40em);
	\filldraw [fill=cyan, draw=black, thick] (5em,10em) circle [radius=4em];
	\filldraw [fill=yellow, draw=black, thick] (5em,0em) circle [radius=4em];
	\node at (5em,2.5em) {Target cell};
	\node at (5em,7.5em) {T-cell};
	\node [rectangle,draw,fill=white,minimum width=5em] at (5em,4.15em) {{\small $010011$}};
	\node [rectangle,draw,fill=white,minimum width=5em] at (5em,5.75em) {{\small $101100$}};
\end{scope}
\end{tikzpicture}
\end{center}

\Glspl{point mutation} being enough for transcription errors to result in different specificities seems to be the only justification given for this treatment. That, and the fact, as \citet{Dasgupta2008} put it: ``[i]n theory, any matching rule defined on a high-level representation can be expressed as a binary matching rule''. Molecules do bind with other molecules with whom complementarity is not pefect --- an \gls{affinity} function is used to test the probability of success for a given molecular binding. There is less grounding and more dispute on the affinity functions. But in general, the probability function is based on the Hamming distance between the two binary strings. For instance, $\text{P}(s_1,s_2)=\exp(-\text{HammingDistance}(s_1,s_2))$.





\section{Cytokines (distributed communication)}
\label{cytokines}

Cytokines are a broad category of small proteins that are used in cell signaling. Cytokines include chemokines, interferons, interleukins, lymphokines, tumour necrosis factor. These are molecules released by cells to communicate. They are released mainly by immune cells, but also by other cell, when damaged for instance.

Chemokines induce movement, directing chemotaxis in nearby responsive cells. Chemotaxis refer to the gradients that guide movement along the environment; chemotaxis gradients may be either positive or negative.

\section{Maturation (distributed learning)}

Babies are born with little protection against disease, since their \glspl{white blood cell} have not yet mature. However they are not completely defenseless. It is thought that mother's milk contains antibodies produced by the mother. The immune system of the mother is compelled to produce antibodies through her contact with the baby, namely kissing.


\section{Lymph nodes}
\label{lymph node}

For our case study, we are mainly concerned on the function of the secondary lymphoid organs known as lymph nodes. These nodes complements the primary organs (the thymus and the bone marrow seen in section \ref{thymus}) These are nodes strategically located along the body, in places next to possible antigen invasions. Examples of these nodes are the spleen (blood infections), the appendix, ```Peyer's patches''' in the intestine (to defend against food with parasites), tonsils and others (because of respiratory influx). These is where usually antigen is presented, and T and B cells are activated.

\begin{figure}[htb]
\centering
\includegraphics[width=30em]{chp1/lymphnode.jpg}
\caption{Anatomy of the lymph node.}
\end{figure}

Conventional lymph nodes are made of loose networks of follicular dendritic cells (not to confuse with the other sentinel dendritic cell) and they are filled with B cells. In case of disease, lymph nodes tend to swell as B cells proliferate in germinative centers.  Follicular dendritic cells, due to their location, also capture opsonized antigens, especially because they have receptors that bind to antibodies, and thus help keeping B cells in the germinative centers alive and producing antibodies. B cells multiply in germinative centers and can double every 6 hours.

\begin{wrapfigure}{r}{0.4\textwidth}
\includegraphics[width=0.4\textwidth]{chp1/lymphRN.jpg}
\caption{Reticulin stain of collagen fibers in the cortex of rat mesenteric lymph node. ‘f’ indicates a B cell follicle. \citep{Kaldjian2001}}
\end{wrapfigure}

500 billion lymphocytes circulate each day through the various lymph nodes. 10,000 lymphocytes exit the blood and enter the average lymph node every second by passing between \textbf{high endothelial cells (HEV)}. On the other hand, \Glspl{dendritic} . Helper T cells in particular recirculate the blood stream in cycles of 12 to 24 hours, helping other lymph nodes. They tend to accumulate in the \textbf{paracortex} (also known as the T cell area), being retained there by adhesion molecules --- which makes sense since dendritic cells are also found in the paracortex. Within the cortex is a reticular network (RN) --- a system of collagen fibers and extracellular matrix (ECM) --- produced and wrapped by fibroblastic reticular cells (FRC); the space between reticular fibers is \SIrange{5}{20}{\micro\meter}, accommodating two or three lymphocytes. While migration velocity of lymphocytes is \SIrange{12}{28}{\micro\meter\per\minute}, it will also depend on the density of the fiber. \citep{Kaldjian2001}

\Glspl{dendritic} arrive at the subcapsular sinus (SCS) through afferent lymph, carrying information from the peripheral tissues, and then migrate into the paracortex to interact with T cells. Several macrophage subsets are distributed at the medulla and SCS regions, where lymphocytes are relatively sparse. While these structures provide lymphocytes enough room for mobility, their exact role is elusive. \citep{Katakai2004}

Naive B cells upon entering the lymph node express receptors for the CXCL13 cytokine produced by follicular \glspl{dendritic} in the areas where opsonized antigen is being presented. Upon finding its cognate antigen, B cells downregulate the expression of CXCL13 receptors and upregulate the expression of CCR7 receptors. These receptors detects the cytokine that is produced by cells in the border region where activated helper T cells and B cells meet. Likewise, helper T cells which have been activated by dendritic cells also upregulate the expression of CCR7 receptors --- afterwards, some T cells lose the CCR7 and upregulate a receptor that guides them into the B cell follicles to help with class switching and somatic hypermutation.

HIV free virions live primarily in the lymph node. It is estimated that the FDC-associated pool of HIV RNA is about $10^{11}$ copies in a 70kg HIV-infected individual; FDC pool RNA is greater than plasma RNA in the orders of $10^2$ to $10^4$, sometimes a few orders higher. \citep{Haase1996} The effectiveness of differing treatments has also been found to differ for FDC-associated virions. \citep{Hlavacek2000}



\section{HIV}
\label{hiv}

AIDS was first clinically observed in 1981, and was named in 1982. HIV was named in 1986. Genetically, it is clear HIV comes from a mutation of a virus from the simian immunodeficiency virus (SIV) family which often infects humans that eat wild chimpanzees, but is inoffensive. According to a \citet{unaids} report, approximately 35.3 million people are reportedly living with HIV worldwide.

An HIV free virion is an enveloped RNA virus of \SI{120}{\nano\meter} of diameter (in average) \citep{Zhu2003}, whose outer surface bears the glycoprotein
gp120, containing several enzymes besides the RNA. Free virions infect target cells by the binding of gpl20 to CD4 and a chemokine coreceptor on the target cell surface, as discussed in section \ref{macrophage}.
[cite Involvement of macrophage mannose]
It is much smaller than \glspl{white blood cell}, and are produced at massive quantities; in any given blood sample, they are present in quantities several orders higher than those of \glspl{white blood cell}. In fact, usual HIV tests only measure free virion RNA copies, and T cells counts are usually not easily measured. This order of magnitudes makes a challenge for agent-based and other fine-grained simulations. Like most authors, we only consider HIV-1 which is widespread in the West, and not HIV-2 which is endemic in West Africa.

The virus contains the following \glspl{enzyme} which will be elaborated later over the section \ref{treatment} on HIV treatments:

\vspace{-\topsep}
\begin{itemize}
\item \gls{reverse transcriptase} enzyme
\item protease enzyme
\item integrase enzyme
\end{itemize}
\vspace{-\topsep}

HIV dynamics are similar to those of any other T-cell dependent disease. It is properly detected and the immune system is activated as expected. Symptoms are identical to those of the flu, as the immune system broadcasts \glspl{cytokine} to prepare all the players for battle.

\tikzstyle{cell} = [draw, ellipse, node distance=4.2ex and 4em]
\tikzstyle{transition} = [font=\small]
\tikzstyle{line} = [draw, thick, ->]

\begin{figure}[htb]
\centering
\begin{tikzpicture}
\node [cell] (antigen) {antigen};
\node [cell, right=of antigen, align=center] (apc) {APC\\\footnotesize{e.g. dendritic cell}};
\node [cell, right=of apc] (th) {T\textsubscript{H}};
\node [cell, right=3em of th] (cyto) {cytokines};
\node [cell, below=of th] (b) {bcell};
\node [cell, left=of b] (plasma) {Plasma};
\node [cell, left=of plasma] (ab) {antibodies};
\node [cell, above=of th] (tc) {T\textsubscript{C}};
\node [cell, left=10em of tc] (infected) {infected T\textsubscript{H}};

\path [line] (antigen) -- (apc) node [transition,pos=0.5,above] {captured};
\path [line] (apc) -- (th) node [transition,pos=0.5,above] {presents to};
\path [line] (th) -- (cyto) node [transition,pos=0.5,above] {produces};
\path [line] (cyto) -- (b) node [transition,pos=0.5,left] {co-stimulates};
\path [line] (b) -- (plasma) node [transition,pos=0.5,above] {produces};
\path [line] (plasma) -- (ab) node [transition,pos=0.5,above] {produces};
\path [line] (ab) -- (antigen) node [transition,pos=0.5,right] {kills};
\path [line] (th) -- (tc) node [transition,pos=0.5,right] {activates};
\path [line] (apc) -- (tc);
\path [line] (tc) -- (infected) node [transition,pos=0.5,above] {kills};
\path [line] (antigen) -- (infected) node [transition,pos=0.5,right] {infects};
\end{tikzpicture}
\caption{HIV immune response overview.}
\end{figure}


The disease starts when the virus penetrates the rectal or vaginal mucosa infecting CD4\textsuperscript{+} cells. As the virus infects helper T cells, the reverse transcription enzyme it carries converts the RNA it carries into DNA. Another enzyme it carries (protease) cuts cell's DNA, allowing viral DNA to merge with the cell's. The cell is therefore instructed to produce HIV copies --- until it bursts (though this is disputed as discussed previously). It is thought there is a latency period, during which no HIV copy is produced, and during which the infection is not detected by killer T cells. Only later, in response to unknown signals, it is thought the virus starts being produced. Furthermore, HIV's reverse transcription enzyme is very error-prone and due to rapid mutations, the new strain may not be recognized as well by killer T cells and antibodies, that were trained and negatively selected for the old strain. According to this theory, HIV dominates the immune system mainly because:

\begin{enumerate}
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item the virus has a high replication rate --- this is why it is never fully cleared;
\item it can hide as a provirus, where it goes undetected;
\item it mutates too rapidly, forcing the adaptive system to go through the process of fighting a new virus, all the while attacking a crucial component of the system.
\end{enumerate}

Another popular school of thought --- immune network theory --- says HIV manages to mutate in such a way as to have its epitopes resemble too closely the V regions of the main population of killer T cells. This means, B cells come to produce antibodies that kill the immune system's own killer T cells. Vaccines are in the works based on this theory.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
Infected productive T life & \SI{2.2}{\day} & \citep{Perelson1996} \\
\hspace{1em} half life & \SI{1.6}{\day} & \citep{Perelson1996} \\
Production & \SI{10.3e9}{virions\per day} & \citep{Perelson1996} \\
Plasma virions avg life & \SI{0.3}{\day} & \citep{Perelson1996} \\
\hspace{1em} half life & \SI{0.24}{\day} & \citep{Perelson1996} \\
Min in-vivo life cycle & \SI{1.2}{\day} & \citep{Perelson1996} \\
Release to next generation & \SI{2.6}{\day} & \citep{Perelson1996} \\
\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}



\subsection{HIV treatments}
\label{treatment}

Several treatments in the form of drugs exist. In fact, treatments have been so successful that HIV has in some parts of the world become a chronic disease. The first drug (AZT) was approved in 1987 --- the fastest drug ever to be approved due to the panic of the booming disease --- and since then about other 30 drugs have been approved. Due to the high mutation rate of the virus --- rather than using one drug at a time --- a cocktail of drugs is used so that it is very much less likely the virus mutates resistance against all drugs at once. This technique is called Highly Active Antiretroviral Therapy (HAART) and it combines several antiretrovirals, sometimes in the same pill. Those antiretrovirals include:

\vspace{-\topsep}
\begin{itemize}
\item \textbf{Nucleotide Reverse Transcriptase Inhibitors (NRTIs, aka ``nukes'')} and \textbf{Non-Nucleoside Reverse Transcriptase Inhibitors (NNRTIs, aka ``non-nukes'')} both inhibit the reverse transcriptase enzyme, which converts viral RNA into DNA --- nukes act in the genetic material, while non-nukes act directly on the enzyme. It is a distinction without a difference from the point of view of intercellular modeling. The first drugs (such as AZT) were reverse transcriptase inhibitors. NRTIs can interfere with mitochondrial DNA, while NNRTIs have neuro-psychiatric effects including suicidal ideation.
\item \textbf{Protease inhibitors (PIs)} inhibit the protease enzyme, which is used to cut cell's DNA.
\item \textbf{Integrase inhibitors (INSTIs)} inhibit the integrase enzyme, which is used to integrate viral DNA into the cell's DNA. They are recent, but seem the best tolerated.
\item \textbf{Entry or fusion inhibitors} target CD4\textsuperscript{+} receptors which HIV uses to infect the cell. Maraviroc targets helper T cells CCR5 receptor, and, in some case, HIV mutates to target the CXCR4 receptor. (In fact, individuals with a naturally faulting CCR5 receptor resist better to the progression of AIDS. The only known adult case of HIV cure was due to a bone marrow transplant from a donor's lacking the CCR5 receptor.) Enfuvirtide is a peptide drug that acts on HIV's gp41.
\end{itemize}
\vspace{-\topsep}

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
\hiderowcolors
ABT-538 PI & & \\
HIV mean half-life & \SI{2.1 +- 0.4}{\day} & \citep{Ho1995} \\
\showrowcolors
\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}








\subsection{HIV dynamics in time}

It is thought HIV-1 and HIV-2 have their origins in a mutation of a simian immunodeficiency virus (SIV) which affects chimps and often infects humans (when eating their meat) but is not long-lived. AIDS (the syndrome) was first named in 1982, and HIV (the cause) was first named in 1986. It is thought that around 35,3 million people around the globe are infected by HIV, as of 2012. \citep{unaids}

HIV infection is characterized by \textbf{three phases}. \textbf{Acute phase.} For the first 2-6 weeks after HIV enters the host, patients experience flu-like symptoms. Then, during the 9th to the 12th week, viral load experiences a sharp decline and the number of helper T cells returns almost to normal levels. \textbf{Chronic phase} or \textbf{clinical latency.} The second phase consists in a long asymptomatic period, which may last for 1 to 10 years, or even more. During this phase viral load is low, but the population of helper T cells keep dropping until a critical point. It is not known exactly what happens during this stage; we see bursts in the data, so some think HIV is incubated in helper T cells until it destroys them and is released; while others think the bursts in the data can be reproduced by continous reproducing using stochastic models. \textbf{Profound immuno-suppression.} In the final phase, the impaired immune system no longer can fight infections and opportunistic diseases take over. The patient is considered to have AIDS. This is declared when \glspl{thcell} count drops below \SI{200}{cells\per\micro\liter} and the viral load increase beyond the \SI{125}{copies\per\micro\liter}.

\begin{figure}[htb]
\centering
\includegraphics[width=34em]{chp1/aids3phases.jpg}
\caption{Three-stages representation of AIDS.}
\end{figure}





\subsection{HIV dynamics in space}
\label{space dynamics}

HIV has different dynamics in different parts of the body. For instance, the clearing rate of the virus (the rate at which it is destroyed) is higher when in the blood (300 day\textsuperscript{-1}) than when in the \gls{lymph node} (5.5 day\textsuperscript{-1}). The virus is most plentiful in \glspl{lymph node}, the spleen, and other organs rich in CD4\textsuperscript{+} cells. Models should take this into account. \citep{Graw2012}

It is less known about the movement of HIV molecules. We do know it is independent of chemokine gradients, unlike the other immune cells --- a wide range of approaches has been employed to study this problem, as surveyed in the following chapter.

As a first approximation of the disease it is interesting to divide the model by compartments with different viral dynamics --- namely, to compensate differing rates of viral replication and clearance. \citep{Graw2012} The virus can either spread through free transmission through the body, subjecting itself to humoral mediated neutralization (antibodies), opsonization and phagocytosis; or they can transmit through cell-to-cell contact avoiding exposure to plasma components, but limiting its scope to the neighborhood of local target cells, spreading like a wave. Cell-to-cell transmission has been shown to be several fold more efficient than melee transmission. This may explain why HIV can be found in areas densely populated with appropriate target cells, such as in the \gls{lymph node}.

\tikzstyle{organ} = [draw, node distance=3em, minimum width=4em]
\tikzstyle{line2} = [draw, thick, <->]

\begin{figure}[htb]
\centering
\begin{tikzpicture}
\node [organ] (lymph) {\gls{lymph node}};
\node [organ, right=of lymph] (blood) {blood};
\node [organ, right=of blood] (lung) {lung};
\node [organ, below right=4ex and 0em of blood] (liver) {liver};

\path [line2] (lymph) -- (blood);
\path [line2] (blood) -- (lung);
\path [line2] (blood) -- (liver);
\path [line2] (lung) -- (liver);
\end{tikzpicture}
\caption{Compartments of differing HIV dynamics.}
\end{figure}
\smallskip

Motility of immune cells is of utmost importance for HIV dynamics. Some studies have been performed on immune cell motility using photon imaging techniques. T cells move in random walks within the \gls{lymph node}. \citet{Miller2003} finds this is so even when considering the entire 3D space, while \citet{Miller2004b} finds T cells are not attracted to \glspl{dendritic} by chemotactic gradients but rather encounter them by chance. \citet{Miller2002}, by harvesting \glspl{lymph node} from mice and performing two-photon imaging on them, has found that the mean absolute displacement of T and B cells from their origin increases proportionally with the square root of time, indicating that both cell types use a random-walk search strategy within the lymph node. However, when challenged by an antigen, they move in ``swarms'' in stable clusters. \citep{Miller2002} Cells move by deforming their shapes; the receptors that are stimulated ``push'' into that direction, provoking an elongation of the cell into that direction, and eventually movement. \citet{Miller2002} furthermore has found that T cells (on mice) when stimulated can either form tight clusters which are relatively stationary, while others formed clusters that roamed more freely massed in ``swarms'' looping within regions a few tens of micrometers across. On the mice, stationary cells (velocity \SI{<2}{\micro\meter\per\minute}) increased from 3\% to 25\% when activated/primed, and the diameter of this clusters were \(~1.5\) times than the unprimed ones. These clusters were contributed mostly by priming and dividing cells --- clusters of cells that have divided three or more times were found to have 21\% of stationarity cells rather than 10\% of the others --- but the velocity of motile cells seemed in average identical irrespective of number of divisions. Freely motile cells that encountered one of these clusters would usually get stuck, but sometimes dissociated and moved freely again.

T cells encounter \glspl{dendritic} randomly, decelerating only slightly while in contact with \glspl{dendritic}, and quickly migrate away after $\sim$\SI{3}{\minute}, while immunized cells seem to suffer no motility drop back. Furthermore, \citet{Miller2004} finds dynamics depend on much on the timescale of the interaction. Before the \SI{2}{\hour}, it seems encounters results in T cells moving in loops, making serial encounters with the same \gls{dendritic} or any neighboring \glspl{dendritic} --- as a result, motility decreases (mean velocity = \SI{5.4}{\micro\meter\per\minute} and motility coefficient = \SI{9.7}{\micro\meter\squared\per\minute}), and interactions were found to be more prolonged when in the presence of antigen (mean = \SI{11.4}{\minute} for the unimmunized mice). In the next stage, \SIrange{2}{14}{\hour}, T cells remained associated with \glspl{dendritic} for longer ($\sim$\SI{60}{\minute}) --- interactions were either interrupted by the T cells moving away or the \glspl{dendritic} withdrawing their dendrites. Entire clusters of T cells transferred sometimes between \glspl{dendritic}. During this period, average T cell velocity was \SI{2.6}{\micro\meter\per\minute}, and the motility coefficient was \SI{2.3}{\micro\meter\squared\per\minute}. Interactions within \SIrange{16}{24}{\hour} involved interactions of an average \SI{20}{\minute}, while T cells looped in ``swarms'' at velocities of \SI{4.1}{\micro\meter\per\minute} and motility coefficient of \SI{6.3}{\micro\meter\squared\per\minute}, although some T cells remained stably associated with its \glspl{dendritic}. After the \SI{24}{\hour}, T cell swarming diminished and many T cells migrated autonomously making infrequent and brief contacts of mean $\sim$\SI{12}{\minute} --- overall T cell velocity was of \SI{4.6}{\micro\meter\per\minute}, while for individual blasts was of \SIrange{8}{9}{\micro\meter\per\minute}, and many instances of cell division were observed; daughter cells regained motility rapidly. Finally, by \SI{40}{\hour} most T cells had undergone one or more rounds of division. \citet{Mempel2004} has divided the dynamics in 3 stages only.

\begin{center}
\begin{tabularx}{0.85\linewidth}{lllll}
\textbf{Stage} & \textbf{Time range} & \textbf{Interaction time} & \textbf{Avg velocity} & \textbf{Motility coefficient} \\\midrule
Stage I & \SI{<2}{\hour} & \SI{11.4}{\minute} & \SI{5.4}{\micro\meter\per\minute} & \SI{9.7}{\micro\meter\squared\per\minute} \\
Stage II & \SIrange{2}{14}{\hour} & \SI{>60}{\minute} & \SI{2.6}{\micro\meter\per\minute} & \SI{2.3}{\micro\meter\squared\per\minute} \\
Stage III & \SIrange{16}{24}{\hour} & \SI{20}{\minute} & \SI{4.1}{\micro\meter\per\minute} & \SI{6.3}{\micro\meter\squared\per\minute} \\
Stage IV & \SI{>24}{\hour} & \SI{12}{\minute} & \SI{4.6}{\micro\meter\per\minute} & \\
\hline
\end{tabularx}
\end{center}

Motility coefficients are calculated assuming random walks as: \(M = x^2 / 4t\).

For our study, the most interesting regions of the \gls{lymph node} are primary follicles (PF), diffuse cortex (DC), medulla (M), afferent lymphatics (AL), and efferent lymphatics (EL). T cells can be found uniquely in the DC, while B cells are found mostly in the PF. The \gls{lymph node} is compartmentalized, and the fluid flows are isolated from the bulk flow of the lymph or blood. Movement of cells is directed by chemokine gradients, whose cells' receptors respond to, and structural pathways of the organ.

HIV treatment also is influenced by spatiality, though this is less often modeled.

A further complication is HIV locomotion within the infected cell. Because of the high viscosity of the cytoplasm, it is unlikely diffusion alone suffices for the reach of the nucleus. Different viruses employ different strategies. \citet{McDonald2002} uses microscopic analysis to argue HIV uses cytoplasmic dynein (a molecular motor) to move towards the nucleus of the cell in curvilinear paths.

\noindent
\rowcolors{1}{}{gray!10}
\begin{center}
\begin{tabularx}{0.85\linewidth}{lXl}
\textbf{Property} & \textbf{Value} & \textbf{Source} \\\midrule
T cell mean velocity & \SI{10.8 +- 0.1}{\micro\meter \per \minute} & \citep{Miller2002} \\
T cell distribution & ``broader'' and ``skewed'' with velocities observed \SI{>25}{\micro\meter \per \minute} & \citep{Miller2002} \\
B cell mean velocity & \SI{6.4 +- 0.07}{\micro\meter \per \minute} & \citep{Miller2002} \\
\hline
\end{tabularx}
\end{center}
\rowcolors{1}{}{}








\subsection{Dynamics of HIV space of strains}

Another kind of space dynamics that has been more intensively studied is with regard to the evolution of HIV strains, oftentimes referred as the ``shape space'' of HIV, as originally studied by \citet{Perelson1979}.



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